(I) In a ballistic pendulum experiment, projectile 1 results in a maximum height h of the pendulum equal to 2.6 cm. A second projectile (of the same mass) causes the pendulum to swing twice as high, h₂ = 5.2 cm . The second projectile was how many times faster than the first?

Verified step by step guidance

1. Identify the relationship between the height reached by the pendulum and the kinetic energy of the projectile. The potential energy at the maximum height (mgh) is equal to the kinetic energy transferred to the pendulum (1/2 mv^2).
2. Set up the equation for the potential energy at the maximum height for both projectiles. For the first projectile: mgh₁ = 1/2 mv₁^2 and for the second projectile: mgh₂ = 1/2 mv₂^2.
3. Since the masses (m) and the gravitational constant (g) are the same for both projectiles, you can simplify the equations to h₁ = 1/2 v₁^2 and h₂ = 1/2 v₂^2.
4. Divide the equation for the second projectile by the equation for the first projectile to find the ratio of the squares of the velocities: (h₂ / h₁) = (v₂^2 / v₁^2).
5. Take the square root of both sides of the equation to solve for the ratio of the velocities: v₂ / v₁ = sqrt(h₂ / h₁). Substitute the given values of h₁ and h₂ to find the ratio of the velocities.

Key Concepts

Conservation of Momentum

In a ballistic pendulum experiment, the conservation of momentum principle states that the total momentum before and after the collision remains constant, provided no external forces act on the system. This principle is crucial for analyzing the initial velocity of the projectile and the resulting motion of the pendulum after impact.

Potential Energy and Maximum Height

The potential energy of an object at a height is given by the formula PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height. In this experiment, the maximum height reached by the pendulum after being struck by the projectile is directly related to the kinetic energy transferred from the projectile, which can be used to compare the speeds of the two projectiles.

Kinetic Energy and Velocity

Kinetic energy (KE) is defined as KE = 0.5mv², where m is mass and v is velocity. The relationship between kinetic energy and velocity is quadratic, meaning that if the kinetic energy changes, the velocity changes by the square root of that change. This concept is essential for determining how many times faster the second projectile was compared to the first, based on the heights achieved.